An electromechanical resonator utilizing flexural vibration has been notified in recent years.
An example of electromechanical resonators of the related art will be explained with reference to FIG. 18. FIG. 18 is a diagram simply showing the configuration of a mechanical vibration filter utilizing the flexural vibration, which is disclosed in a non-patent document 1.
This filter is formed by forming a pattern on a silicon substrate using a thin-film forming process. The filter is configured by an input line 104, an output line 105, both-end-supported beams 101, 102 disposed so as to oppose to the input and output lines with a space of 1 micron or less therebetween, respectively, and a coupling beam 103 for coupling these two both-end-supported beams. A signal inputted from the input line 104 performs capacitive coupling with the beam 101 to generate electrostatic force at the beam 101. The mechanical vibration arise only when the signal frequency is near the resonance frequency of an elastic structure formed by the beams 101, 102 and the coupling beam 103. The mechanical vibration is further detected as the change of an electrostatic capacitance between the output line 105 and the beam 102 thereby to extract the filtrating output of the input signal.
In the case of a both-end-supported beam with a rectangular sectional shape, the resonance frequency f of the flexural vibration will be represented by the following expression supposing that a symbol E denotes an elasticity modulus, p denotes a density, h denotes a thickness and L denotes a length.f=1.03 h/L2·√E/ρ  [Expression 1]
When the material is polycrystalline silicon, the elasticity modulus E is 160 GPa and the density ρ is 2.2×103 kg/m3. Further, when the size is that L=40 μm and h=1.5 μm, the resonance frequency f is 8.2 MHz, whereby a filter with about 8 MHz band can be configured. When the mechanical resonance is employed, it is possible to obtain a sharp frequency selection characteristics with a high Q value as compared with a filter configured by a passive circuit such as a capacitor and a coil.
However, in the aforesaid configuration, there are the following restrictions in order to configure a filter with a higher frequency band. That is, as clear from the expression 1, firstly it is desirable to increase E/ρ by changing the material. However, when E is made larger, a displacement value of the beam becomes smaller even when a force for bending the beam is the same, whereby it becomes difficult to detect the displacement of the beam.
Further, when an index representing the degree of ease of bending the beam is expressed by a ratio d/L between a bending amount d at the center of the beam when applying a static load to the surface of the both-end-supported beam and the length L of the beam, the ratio d/L is expressed by the following proportional expression.d/L is proportional to L3/h3·1/E  [Expression 2]
From this expression, in order to raise the resonance frequency while maintaining the value d/L, since at least E can not be changed, it is necessary to obtain the material with a low density ρ. Thus, as the material having almost the same Young's modulus as polycrystalline silicon and a low density, it is necessary to use composite material such as CFRP (Carbon Fiber Reinforced Plastics). In this case, it is difficult to constitute a fine mechanical vibration filter by the semiconductor process.
Thus, as a second method of not using such the composite material, there is a method of changing the size of the beam in the expression 1 to increase h/L2. However, when the thickness h of the beam is made larger and the length L of the beam is made smaller, the index d/L of the expression 2 representing the degree of ease of bending becomes smaller and so it becomes difficult to detect the bending of the beam.
FIG. 19 shows a relation between log(L) and log(h) in the expressions 1 and 2, in which a straight line 191 shows a relation obtained from the expression 1 and a straight line 192 shows a relation obtained from the expression 2. In FIG. 19, when values L and h in a range (area A) above the straight line of the inclination “2” which starts from the current size point A are selected, the value f becomes larger. In contrast, when values L and h in a range (area B) beneath the straight line of the inclination “1” are selected, the value d/L becomes larger. Thus, a hatched portion (area C) in the drawing represents an area of the values L and h which can increase the resonance frequency while securing the bending amount of the beam.
It is clear from FIG. 19 that it is a necessary condition to fine both the length L and the thickness h of the beam in order to realize the high-frequency characteristics of the mechanical vibration filter. Further, it a sufficient condition shown by the hatched portion in FIG. 19 to reduce the values L and h with the same scaling, that is, to reduce the values L and h along the straight line of the inclination “1”.
In this manner, in the mechanical resonator of the related art, the resonance frequency can be made high by miniaturizing the size of the mechanical vibrator. However, in general, there is a problem that the mechanical Q value of the flexural vibration reduces when the size is miniaturized. As to this phenomenon, a non-patent document 2 shows a result in which the relation among a length and a thickness of a beam and a Q value of flexural resonance is measured by using a cantilever of mono-crystalline silicon. This non-patent document 2 shows that the Q value reduces by shortening the length of the beam and reducing the thickness of the beam. Thus, when the resonator using the flexural vibration according to the related art is miniaturized and applied to a filter, there arises a problem that a Q value necessary for obtaining the desired frequency selection characteristics may not be obtained.    Non-patent document 1: Frank D. Bannon III, John R. Clark, and Clark T-C. Nguyen, “High-QHF Microelectromechanical Filters”, IEEE Journal of Solid-State Circuits, Vol. 35, No. 4, pp. 512-526, April 2000.    Non-patent document 2: K. Y Yasumura et al., “Quality Factors in Micron-and Submicron-Thick Cantilevers”, IEEE Journal of Microelectromechanical Systems, Vol. 9, No. 1, March 2000.